Differentiating the single-layer potential

Question

Suppose $f\in L^2[-1,1]$ and consider the single layer potential with moment $f$ on $[-1,1]$ $$ Kf(x,y) = -\frac{1}{2\pi}\int_{-1}^1 \ln|(x,y) - (\xi,0)|f(\xi)\, d\xi $$ Formally I shown that for $x\in[-1,1]$ $$ \frac{\partial Kf}{\partial y}\bigg|_{y=0} = \frac{f(x)}{2} $$ by differentiating under the integral sign. But I have trouble justifying interchanging the differentiation and integration. As far as I am aware, this is doable if $f\in C^0[-1,1]$ (or maybe $f\in C_c^0[-1,1]$), but I'm not entirely sure if I am allowed to simply abuse the fact that $C^0[-1,1]$ is dense in $L^2[-1,1]$. I am looking for a reference to this result if possible, since layer potential theory is well-studied.